Question on gravitation?
Two small spaceships, each with mass m = 2000 kg, are in the circular Earth orbit of the figure, at an altitude h of 440 km. Igor, the commander of one of the ships, arrives at any fixed point in the orbit 84.0 s ahead of Picard, the commander of the other ship. What are the (a) period T0 and (b) speed v0 of the ships? At point P in the figure, Picard fires an instantaneous burst in the forward direction, reducing his ship's speed by 1.30%. After this burst, he follows the elliptical orbit shown dashed in the figure. What are the (c) kinetic energy and (d) potential energy of his ship immediately after the burst? In Picard’s new elliptical orbit, what are (e) the total energy E, (f) the semimajor axis a, and (g) the orbital period T? (h) How much earlier than Igor will Picard return to P?
Two small spaceships, each with mass m = 2000 kg, are in the circular Earth orbit of the figure, at an altitude h of 440 km. Igor, the commander of one of the ships, arrives at any fixed point in the orbit 84.0 s ahead of Picard, the commander of the other ship. What are the (a) period T0 and (b) speed v0 of the ships? At point P in the figure, Picard fires an instantaneous burst in the forward direction, reducing his ship's speed by 1.30%. After this burst, he follows the elliptical orbit shown dashed in the figure. What are the (c) kinetic energy and (d) potential energy of his ship immediately after the burst? In Picard’s new elliptical orbit, what are (e) the total energy E, (f) the semimajor axis a, and (g) the orbital period T? (h) How much earlier than Igor will Picard return to P?
1 Answer
(a) For object of mass
T_0^2=(4pi^2)/(GM)r^3 ......(1)
whereG is Universal Gravitational Constant.
In terms of altitude of the spaceships
T_0=sqrt((4pi^2)/(GM)(R+h)^3)
Inserting various values we get
(b) Centripetal force is balanced by gravitational force. The expression becomes
(mv_0^2)/r=(GMm)/r^2
=>v_0=sqrt((GM)/r)
Alternately, for a circular orbit
v_0=romega
=>v_0=(R+h)(2pi)/T_0
Inserting various values in alternate expression
v_0=(6.81xx10^6)(2pi)/5591
=>v_0=7653\ m cdot s^-1
(c) Kinetic energy of Picard's spaceship just after firing of burst
E_K=1/2mv^2
Inserting various values we get
E_K=1/2(2000)((100-1.30)/100xx7653)^2
=>E_K=5.7xx10^10\ J
(d) Potential energy of this spaceship at the same time
E_P=-(GMm)/(r)
Inserting various values we get
E_P=-((6.67xx10^-11)(5.98xx10^24)(2000))/(6.81xx10^6)
E_P=-1.17xx10^11\ J
(e) Total energy
E_T=-1.17xx10^11+5.7xx10^10
E_T=-6.0xx10^10\ J
(f) Semi major
E_T=-(GMm)/(2a)
=>a=-(GMm)/(2E_T)
Inserting given values we get
=>a=-((6.67xx10^-11)(5.98xx10^24)(2000))/(2(-6.0xx10^10))
=>a=6.65xx10^6\ m
(g) The new orbital period
T^2=(4pi^2)/((6.67xx10^-11)(5.98xx10^24))(6.65xx10^6)^3
=>T=sqrt((4pi^2)/((6.67xx10^-11)(5.98xx10^24))(6.65xx10^6)^3)
=>T=5395.1\ s
(h) Picard is now faster than Igor by time
DeltaT=5591.0-5395.1=195.9\ s
When he arrived first at point
195.9-84.0=111.9\ s